On the asymptotic optimum allocation in estimating inequality from complete data. (English) Zbl 0771.62083

Summary: Studies dealing with the quantification of inequality of a population with respect to a given quantitative attribute, provide us with a large class of measures. Among these, we can distinguish, because of their properties and operativeness, the ones coinciding with, or being ordinally equivalent to, the dimensionless “additively decomposable inequality indices”.
As indicated in previous papers, many populations, whose inequality in relation with an attribute is useful to quantify, are too large to be censused but large samples from them can be drawn and they arise naturally stratified. On the basis of these last two advantages, we will approach the optimum allocation in estimating inequality, and a comparison with the proportional allocation, and with the absence of strata, will be later established.


62P20 Applications of statistics to economics
62D05 Sampling theory, sample surveys
91B82 Statistical methods; economic indices and measures
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[1] A. B. Atkinson: On the measurement of inequality. J. Econom. Theory 2 (1970), 131-143.
[2] P. Bickel, K. A. Doksum: Mathematical Statistics. Holden-Day, Inc., Oakland 1977. · Zbl 0403.62001
[3] Y. M. M. Bishop S. E. Fienberg, P.W. Holland: Discrete Multivariate Analysis: Theory and Practice. MIT Press, Cambridge 1975. · Zbl 0332.62039
[4] F. Bourguignon: Decomposable income inequality measures. Econometrica 47 (1979), 901-920. · Zbl 0424.90013
[5] C. Caso, M.A. Gil: Estimating income inequality in the stratified sampling from complete data; Part 1: The unbiased estimation and applications. Kybernetika 25 (1989), 298-311. · Zbl 0682.62086
[6] C. Caso, M.A. Gil: Estimating income inequality in the stratified sampling from complete data; Part 2: The asymptotic behaviour and the choice of sample size. Kybernetika 25 (1989), 312-319. · Zbl 0682.62087
[7] F. A. Cowell: On the structure of additive inequality measures. Rev.of Econom.Stud. 47 (1980), 521-531. · Zbl 0437.90035
[8] F. A. Cowell, K. Kuga: Additivity and the entropy concept. An axiomatic approach to inequality measurement. J. Econom. Theory 25 (1981), 131-143. · Zbl 0487.90039
[9] W. Eichhorn, W. Gehrig: Measurement of inequality in economics. Modern Applied Mathematics - Optimization and Operations Research (B. Korte, North-Holland, Amsterdam 1982. · Zbl 0491.90024
[10] M.A. Gil: A note on the stratification and gain in precision in estimating diversity from large samples. Commun. Statist. - Theory and Methods 18 (1989), 1521-1526. · Zbl 0696.62261
[11] M. A. Gil C. Caso, P. Gil: Estudio asintotico de una clase de indices de desigualdad muestrales. Trab. de Est. 4 (1989), 95-109. · Zbl 0731.62158
[12] M.A. Gil R. Perez, P. Gil: A family of measures of uncertainty involving utilities: definitions, properties, applications and statistical inferences. Metrika 36 (1989), 129-147. · Zbl 0697.62004
[13] I. Horowitz: Employment concentration in the common market: an entropy approach. J. Roy. Statist. Soc Ser. A 133 (1970), 463-679.
[14] C. R. Rao: Linear Statistical Inference and its Applications. Wiley, New York 1973. · Zbl 0256.62002
[15] A.F. Shorrocks: The class of additively decomposable inequality measures. Econometrica 48 (1980), 613-625. · Zbl 0435.90040
[16] H. Theil: Economics and Information Theory. North-Holand, Amsterdam 1967.
[17] D. Zagier: On the decomposability of the Gini coefficient and other indices of inequality. Discussion paper No. 108. Projektgruppe Theoretische Modelle. Universität Bonn, 1983.
[18] J. Zvárová: On asymptotic behaviour of a sample estimator of Rényi’s information of order \(\alpha\). Trans. 6th Prague Conf. on Inf. Theory, Stat. Dec. Func, Rand. Proc, Academia Prague 1973, pp. 919-924. · Zbl 0299.94017
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